Distances and component sizes

in scale-free random graphs

Joost Jorritsma, PhD

Florence Nightingale Bicentennial Fellowship

Research talk

Internet: a growing network of routers and servers

~1969: 2 connected sites

Time

~1989: 0.5 million users

~2023: billions of devices

[Faloutsos, Faloutsos & Faloutsos, '99]:
- Short average distance:
  Quick spread of information

~1999: 248 million users

Distance evolution in a growing network

1999

$\mathrm{dist}_{\color{red}{'99}}(u_{'99}, v_{'99}) = 4$

2005

$\mathrm{dist}_{{\color{red}'05}}(u_{'99}, v_{'99}) = 3$

2023

$\mathrm{dist}_{{\color{red}'23}}(u_{'99}, v_{'99}) = 2$

21 possible networks

Attachment rule:

Favour connecting to high-degree vertices, $\tau$ : tail of power-law degree distribution

2005

$\phantom{\mathrm{dist}_{{\color{red}'05}}(u_{'99}, v_{'99}) = 3}$

Distance evolution: hydrodynamic limit

Theorem [J., Komjáthy, Annals of Applied Probability '22]. Assume $\tau<3$ .
Let $t'=T_t(a):=t\exp\big(\log^a(t)\big)$ for $a\in[0,1]$ , then

$\mathbb{P}\bigg(\sup_{a\in[0,1]} \left| \frac{\mathrm{dist}_{T_t(a)}(U_t, V_t)}{\log\log(t)} - (1-a)\frac{4}{|\log(\tau-2)|}\right|>\varepsilon\bigg)\longrightarrow 0.$

Theorem [J., Komjáthy, Annals of Applied Probability '22]. Assume $\tau<3$ .
Let $t'=T_t(a):=t\exp\big(\log^a(t)\big)$ for $a\in[0,1]$ , then

$\phantom{\mathbb{P}\bigg(}\phantom{\sup_{a\in[0,1]}} \left| \frac{\mathrm{dist}_{T_t(a)}(U_t, V_t)}{\log\log(t)} - (1-a)\frac{4}{|\log(\tau-2)|}\right|\phantom{>\varepsilon\bigg)\phantom{\longrightarrow} 0.}$

Theorem [J., Komjáthy, Annals of Applied Probability '22]. Assume $\tau<3$ .
Let $\phantom{t'=T_t(a):=t\exp\big(\log^a(t)\big)}$ for $\phantom{a\in[0,1]}$ , then

$\phantom{\sup_{a\in[0,1]} \left| \frac{\mathrm{dist}_{T_t(a)}(U_t, V_t)}{\log\log(t)} - (1-a)\frac{4}{|\log(\tau-2)|}\right|\overset{\mathbb{P}}{\longrightarrow} 0.}$

Theorem [J., Komjáthy, Annals of Applied Probability '22]. Assume $\tau<3$ .
Let $t'=T_t(a):=t\exp\big(\log^a(t)\big)$ for $a\in[0,1]$ , then

$\phantom{\mathbb{P}\bigg(}\phantom{\sup_{a\in[0,1]}} \left| \frac{\mathrm{dist}_{T_t(a)}(U_t, V_t)}{\log\log(t)} - \phantom{(1-a)\frac{4}{|\log(\tau-2)|}}\right|\phantom{>\varepsilon\bigg)\longrightarrow 0.}$

```
Dynamics in PAMs.
```

Generalization with edge weights: 
random transmission times

Novelties

```
Fast spreading among influentials;
```

Theorem [J., Komjáthy, Annals of Applied Probability '22]. Assume $\tau<3$ .
Let $t'=T_t(a):=t\exp\big(\log^a(t)\big)$ for $a\in[0,1]$ , then

$\mathbb{P}\bigg(\sup_{a\in[0,1]} \left| \frac{\mathrm{dist}_{T_t(a)}(U_t, V_t)}{\log\log(t)} - (1-a)\frac{4}{|\log(\tau-2)|}\right|>\varepsilon\bigg)\phantom{\longrightarrow 0.}$

Information spreading on random graphs

Distances

Component sizes

Intervention strategies

FINAL SIZE

Intervention strategies

Kernel-based spatial random graphs

Only four parameters

Vertex set

Spatial locations, $\mathbb{Z}^d$ or Poisson point process
Vertex weights
(iid Pareto or constant)

Edge more likely if

Spatially nearby,
High weight

\exp\big(-\Theta(n^\zeta)\big).

\exp\big(-\Theta(n^\zeta)\big).

Theorem [J., Komjáthy, Mitsche, '23+]
We find explicit $\zeta\in[1/2,1)$ , $\theta\in(0,1)$ s.t.

Lower tail: small final size
If enough long edges or no hubs

\mathbb{P}\big(|{\color{blue}\mathcal{C}_n^{(1)}}|/n< \theta-\varepsilon\big)=

\mathbb{P}\big(|{\color{blue}\mathcal{C}_n^{(1)}}|/n< \theta-\varepsilon\big)=

Novelties

Reversed discrepancy: slower upper tail.

Long edges can beat surface tension: any         ;

First (upper) LDP for giant in spatial graph;

\zeta\!\in\![1/2, 1)

\zeta\!\in\![1/2, 1)

  drives also other cluster-size distributions

\zeta

\zeta

What is the influence of long edges and high-degree vertices?

\mathbb{P}\big(|{\color{blue}\mathcal{C}_n^{(1)}}|/n>\theta +\varepsilon\big)= \phantom{\Theta\big(n^{-I(\varepsilon)}\big)}

\mathbb{P}\big(|{\color{blue}\mathcal{C}_n^{(1)}}|/n>\theta +\varepsilon\big)= \phantom{\Theta\big(n^{-I(\varepsilon)}\big)}

Upper tail: large final size
If hubs are present, we find rate funtion $I(\varepsilon)$ :

What is the influence of long edges and high-degree vertices?

\Theta\big(n^{-I(\varepsilon)}\big).

\Theta\big(n^{-I(\varepsilon)}\big).

Surface tension in supercritical nearest-neighbour percolation

\mathbb{P}\big(|{\color{blue}\mathcal{C}_n^{(1)}}|/n<\theta - \varepsilon\big)=

\mathbb{P}\big(|{\color{blue}\mathcal{C}_n^{(1)}}|/n<\theta - \varepsilon\big)=

\exp\big(-\Theta(\sqrt{n})\big)

\exp\big(-\Theta(\sqrt{n})\big)

[Grimmett & Marstrand '90, Kesten & Zhang '90, Pisztora '96]

\mathbb{P}\big(|{\color{blue}\mathcal{C}_n^{(1)}}|/n>\theta + \varepsilon\big)=

\mathbb{P}\big(|{\color{blue}\mathcal{C}_n^{(1)}}|/n>\theta + \varepsilon\big)=

\exp\big(-\Theta(n)\big)

\exp\big(-\Theta(n)\big)

[Lebowitz & Schonmann '88]

Discrepancy: faster upper tail
No precise asymptotics

What is the influence of long edges and high-degree vertices?

Lower tail: small final size

Upper tail: large final size

Kernel-based spatial random graphs

Edge set $\mathcal{E}_\infty$

Symmetric kernel $\kappa(w_u, w_v)$ ,
Long-range parameter $\alpha\in(1,\infty]$ ,
Edge-density $\beta>0$ ,

Connection probability

$\mathbb{P}\big(u\leftrightarrow v\mid \mathcal{V}_\infty\big)=\bigg(\beta\frac{\kappa(w_u, w_v)}{\|x_u-x_v\|^d}\bigg)^\alpha\wedge 1$

Vertex set $\mathcal{V}_\infty$

Spatial locations, either
- Lattice $\mathbb{Z}^d$
- Poisson point process (unit intensity)
Power-law i.i.d. weights $w_v\ge 1$ :
$\mathbb{P}(w_v\ge w)=w^{-(\tau-1)}$ ,

$\mathbb{P}\big(u\leftrightarrow v\mid \mathcal{V}_\infty\big)=\bigg(\beta\frac{\kappa(w_u, w_v)}{\|x_u-x_v\|^d}\bigg)^\alpha$

$\mathbb{P}\big(u\leftrightarrow v\mid \mathcal{V}_\infty\big)=\phantom{\bigg(\beta}\frac{\kappa(w_u, w_v)}{\phantom{\|x_u-x_v\|^d}}\phantom{\bigg)^\alpha\wedge 1}$

$\mathbb{P}\big(u\leftrightarrow v\mid \mathcal{V}_\infty\big)=\phantom{\bigg(\beta}\frac{\kappa(w_u, w_v)}{\|x_u-x_v\|^d}\phantom{\bigg)^\alpha\wedge 1}$

$\mathbb{P}\big(u\leftrightarrow v\mid \mathcal{V}_\infty\big)=\bigg(\phantom{\beta}\frac{\kappa(w_u, w_v)}{\|x_u-x_v\|^d}\bigg)^\alpha\phantom{\wedge 1}$

Components in supercritical graphs

Largest component ${\color{blue}\mathcal{C}_n^{(1)}}$ :
- Linear in box size
- Law of large numbers
- Lower tail large deviations
- Upper tail large deviations

Questions

Component of the origin ${\color{green}\mathcal{C}(0)}$ : $\mathbb{P}\big(k\le |{\color{green}\mathcal{C}(0)}|<\infty\big)=$

Second-largest component: $|{\color{red}\mathcal{C}_n^{(2)}}|=\Theta\big(\phantom{what}\big)$

Lower bounds: cluster-size decay

Aim: Find minimal $\zeta$ s.t.

\mathbb{P}\big(k \le |\mathcal{C}(0)|<\infty\big)\gtrsim\exp\big(-k^\zeta\big);

\mathbb{P}\big(k \le |\mathcal{C}(0)|<\infty\big)\gtrsim\exp\big(-k^\zeta\big);

|\mathcal{C}(0)|\ge k

|\mathcal{C}(0)|\ge k

\Bigg\}

\Bigg\}

\Theta(k)

\Theta(k)

k^\zeta\sim\mathbb{E}\big[|\{v\in\Lambda_k: v\leftrightarrow \Lambda_k^c\}|\big]

k^\zeta\sim\mathbb{E}\big[|\{v\in\Lambda_k: v\leftrightarrow \Lambda_k^c\}|\big]

\mathbb{E}[|\phantom{\square\square}|]\sim\mathbb{E}[|\phantom{\square\square}\leftrightarrow \phantom{\square\square}|]

\mathbb{E}[|\phantom{\square\square}|]\sim\mathbb{E}[|\phantom{\square\square}\leftrightarrow \phantom{\square\square}|]

Aim: Find $\gamma$ s.t.

\sim k^\zeta

\sim k^\zeta

{\color{green}\text{Cluster-size decay}}

{\color{green}\text{Cluster-size decay}}

Upper bounds

\mathbb{P}\big(|{\color{red}2^{\mathrm{nd}}\text{-largest}}|\ge k\big)\lesssim n\exp\big(-k^\zeta\big)

\mathbb{P}\big(|{\color{red}2^{\mathrm{nd}}\text{-largest}}|\ge k\big)\lesssim n\exp\big(-k^\zeta\big)

\mathbb{P}\big(|{\color{blue}\text{largest}}|/n\le \varepsilon \big)\lesssim \phantom{n} \exp\big(-n^\zeta\big)

\mathbb{P}\big(|{\color{blue}\text{largest}}|/n\le \varepsilon \big)\lesssim \phantom{n} \exp\big(-n^\zeta\big)

\text{Law of large numbers}

\text{Law of large numbers}

{\text{\color{blue}Lower tail large deviations}}

{\text{\color{blue}Lower tail large deviations}}

Challenge: Delocalized components

\binom{n}k \sim n^k \gg n\exp\big(k^\zeta\big)

\binom{n}k \sim n^k \gg n\exp\big(k^\zeta\big)

# possibilities for $|{\color{red}2^{\mathrm{nd}}\text{-largest}}|\ge k$

1 . 1

Distances and component sizes in scale-free random graphs Joost Jorritsma, PhD Florence Nightingale Bicentennial Fellowship Research talk

Distances and component sizes

in scale-free random graphs

Random Graphs: Probability, Combinatorics, Physics, Epidemiology

3x Information diffusion in random graphs

Distances

Component sizes

Intervention strategies

Internet: a growing network of routers and servers

Distance evolution in a growing network

Distance evolution

Distance evolution: hydrodynamic limit

Information spreading on random graphs

Distances

Component sizes

Intervention strategies

Intervention strategies

Real networks contain many triangles!

Do real networks look like this?

Kernel-based spatial random graphs

Clusters in supercritical graphs

What is the influence of long edges and high-degree vertices?

What is the influence of long edges and high-degree vertices?

Information spreading on random graphs

Distances

Component sizes

Intervention strategies

Distances and component sizes

in scale-free random graphs

Surface tension in supercritical nearest-neighbour percolation

Kernel-based spatial random graphs

Components in supercritical graphs

Lower bounds: cluster-size decay

Upper bounds

Challenge: Delocalized components

Oxford research presentation

Oxford research presentation

joostjor

Distances and component sizes

in scale-free random graphs

Oxford research presentation

More from joostjor